Five recent book reviews
This is the third in a sequence of the yearly collected anthologies of insightful papers on mathematics. The idea is to bridge the gap between professional mathematicians and a broader public. Papers give an accessible account of major mathematical achievements, discuss philosophy and history of mathematics as well as teaching aspects.
After the editions of 2010 and 2011, Pitici succeeds again in bringing together a wonderful collection of papers on mathematics. The foreword is delivered by the Fields Medal winner David Mumford.
The papers are not mathematical in the the technical sense of theorems, proofs and formulas. They are on mathematics and are accessible to a broad public. Like in the previous collections, the major themes are philosophy, teaching, history and the dichotomy and mutual influences of pure and applied mathematics. The papers were carefully selected from the existing literature. They were not written especially for this collection but only typeset in a uniform way. Except for the introduction in which Pitici surveys the contents and the foreword in which Mumford ponders on the synergy of pure and applied, of abstract and concrete mathematics, which is a recurrent theme in other contributions of this volume.
The 24 papers are relatively short (an average of 12 pages) but nevertheless succeed in addressing an intriguing idea, or an insightful mathematical topic and are just long enough to keep the attention of even a slightly interested reader from the beginning to the end. But who shouldn't be highly interested in questions such as Is mathematic discovered or invented? or Why is there a philosophy of mathematics at all? or Why Math works. The reader can also enjoy an exposition about more advanced mathematical topics such as string theory, the zeta function, the volume of a ball in higher dimensions or the mathematics of music and the music of mathematics. Or he might be interested in more playful applications like the mathematics of dancing, origami weaving, of transformations in photography, or the application of game theory applied to mating and dating. History is represented by contributions on Augustus De Morgan, Jean Bernoulli and Georg Cantor, the history of infinity from Hilbert to Woodin and of the history of the routing problem (see also the review of In pursuit of the travelling salesman in this database) and the concurrence of the history of mathematics and science. As important as the history is for the mathematics of today, for the mathematics of the future it is equally importandt how we teach mathematics to our students now. Hence the importance of the papers on math teaching. For example how do we transfer the different meaning that a variable may have in different environments and how the education of the teacher will be passed on unwittingly to the students, sometimes in a very subtle way. This applies to the teacher's background on applications but also, more generally, on his own philosophical opinion. Anyway, the ultimate guidelines on how to be an optimal teacher of mathematics are yet to be defined.
This is indeed a collection of the most wonderful writings on mathematics that have appeared recently. Not elementary at all and yet accessible to a general audience. Of course this is just the top a a gigantic iceberg, a top that has been selected on the basis of space and copyright limitations. This allows to make a selection of the best among the best that have already been selected as the best by demanding journals such as Science, The mathematical Monthly, and Scientific American. However, these are picked out of a longer short-list of possible candidates. Pitici gives a list of 43 papers that have been considered at some point but didn't make it to this volume. The titles in this list sound as appealing as the ones that were included in this volume, which makes you regret that the book is not thicker than it is.
Similar selections are published on a regular basis. For example the Mathematics and Culture series (edited by Michele Emmer, and published by Springer, reviews in this database Vol.I, Vol.III) which concentrates more on the connection between mathematics and all forms of art. The series What's happening in the mathematical Sciences (edited by Barry Cipra and now by Dana MacKenzie, and published by the AMS) which collects papers that explain important mathematical evolutions or breakthroughs for a broader public. By their focus, these series do not overlap with but are complementing the Best writing on mathematics. It is no waste of time to read all three.
The book gives a list of mathematical techniques applied in modelling in healthcare systems
It is a fact that healthcare systems are among the most relevant tasks for governments due the essential social impact of sanitary policy and the increasing budget experienced along the last decades. The complicated questions arising in this context require more than simple intuition to make good decisions. Complex systems require models with which policymakers may steer decisions in the best direction. As the title suggests, the goal of this book is precisely to show a collection of modelling tools applied in healthcare situations. The authors are members of the Complex Systems Modelling Group at Simon Fraser University (Vancouver, Canada) and, though it includes some other approaches (as psychosocial modelling or management questions) the cornerstone of the book is given in mathematical language. This work does not give a complete and rigorous presentation of mathematical techniques, but a panorama of the main tools and indications used in current advanced modelling in healthcare. Actually, potential readers just require a solid high school level of mathematics to follow the core of the book. The structure of it is organized in sixteen chapters (together with an appendix on computer packages useful in modelling), most of them structured in five sections entitled: “model overview” giving a brief description, “common uses” giving a list of examples that the modelling technique could be used to address, “mathematical details”, “examples” applied in practice and “related reading”. The tools covered are descriptive statistics, regression analysis, game theory, graph theory, Markov models, queuing models, optimizations, etc. Even though the expert immediately realizes that such a wide variety of topics can be only covered in a superficial way, the book may represent a valuable introduction for people interested in the world of modelling in healthcare. A serious construction of a model though one (or several) of these techniques will require a deeper study than that found in this book (for example, with the help of the references provided in the related reading sections), but the mere collection of tools and applications offered in it might represent a useful reference for both academic of professional members.
David E. Pilgrim (1933-2005) was an eminent classicist, linguist and historian of Ancient Science, member of the History of Mathematics Department at Brown University. The autor, Philip J. Davis, met him there and both quickly established a solid friendship and an intellectual interchange. “Ancient loons” is dedicated to the memory of David Pilgrim and its goal is to preserve some of his thoughts and specialized knowledge, as well as to remark the importance of curiosity in the study of History.
The book is a collection of short stories, small anecdotes in the life of some historical characters. More concretely, it focuses on the oddities and singularities of some well-known historical figures, not only in Science, but also in Arts, Politics and Social Sciences. These small tales were called “oddballs” or “Loons” by Pilgrim and constitute the core of the book.
John Napier, Heliogabalus, Priscillian, Siva and Parvati, St. Cuthbert and many others form these “Ancient Loons”, which are usually 5 or 6 pages long, so the approach to each character is just anecdotical. Besides, the Loons jump back and forward in time, so no particular connection is made between them, apart from their excentricity or the originality of their beliefs in their particular era. However, the book shows the fascination for Ancient History, the treasures hidden in original sources and the importance of exploring unusual connections. Although some mathematicians (as Napier) and mathematical thoughts (the relationship between Theology and Mathematics) appear in the text, the level is elementary as the book was written for a general audience.
Here is a practical introduction to constructive approximation theory emphasizing polynomial and rational interpolation and approximation. It's a hands-on approach based on the matlab package chebfun. Told in a narrating style with few formulas but many graphs.
Nick Trefethen has a way of catching the attention of his audience during his enthusiastic lectures or of his readership in his publications. Staying close to the basics of computations, replacing an overload of formulas and abstraction with graphics and examples, makes the message easy to accept and almost obvious. The abstraction of approximation theory with existence proofs without constructive arguments and the crafting of exotic counter examples has created several legends living among scientists who may conceive a method as "inherently bad" while it works perfectly well in almost all possible practical situations. This is beautifully illustrated in an appendix to this book, Trefethen adds a reprint of his paper Six myths of polynomial interpolation and quadrature (Math. Today, 47(4) 184-188, 2012). This will be an eye-opener for many.
The same holds for the rest of this book. One of the myths is that polynomial interpolation is bad because it may not converge and it can be numerically unstable. However, assuming a little bit of smoothness of the function, choosing Chebyshev interpolation points and using barycentric representations, makes polynomial interpolation perfectly acceptable, in general being only a tiny bit away from the best polynomial approximant (in max norm).
So The first 20 chapters of this book convinces the reader of this fact discussing classical results like the Weierstrass theorem, Gibbs and Runge phenomenons, Lebesgue constants, rate of convergence, and associated quadrature formulas. All of this is illustrated with chebfun commands. The book is not about chebfun though, and the reader should obviously be familiar with matlab. The numerical examples and exercises that conclude each chapter are essential to assimilate the material. In fact the book which shows (LaTeX) text interlaced with chebfun commands, is actually produced by matlab's publish from matlab files with the text inserted as comments in between the matlab commands. These files should be downloaded from the book's website. So they provide the text of the book as well as the chebfun commands.
The polynomials and their approximating properties being treated, an application to spectral methods is given in chapter 21 and in chapter 22 a conformal transformation is used as a tool to increase the power of plain polynomial interpolation and approximation. The next 6 chapters switch to rational approximation. But again, Trefethen comes to a somewhat counter intuitive conclusion that in many situations rationals are not really much better than polynomials, except for approximating near singularities or approximation on unbounded intervals. The chapter on Padé approximation is remarkable because it gives a very stable implementation based on SVD, clearly computing the singular blocks in the Padé table and avoiding spurious zeros and poles (the ones that nearly cancel each other).
As a result of the chebfun approach, the whole book is concentrating mainly on real approximation on the interval [-1,1] (Chebyshev polynomials). Using a Joukowski transform x = cos(z), x = exp(iθ) allows to translate everything to the complex unit circle (Laurent / trigonometric polynomials) or the interval [-π,π] (Fourier analysis). However, Trefethen has chosen to stick to the Chebyshev situation for clarity.
The book is an easy read. It is highly recommended both for beginning students in approximation theory and/or numerical analysis and for the knowledgeable researcher. The book should be accompanied with a computer with the chebfun package installed but also with some other (probably more theoretical or classical) book on approximation theory. In this book, the theorems are accompanied with proofs that are usually not very formal, or just indication the method or the road to be followed. It might be advisable to look up details in another book. Most enjoyable are the historical remarks with which Trefethen seasons his discourse. They are most informative and sometimes pleasant anecdotes.
The volume consists of 15 chapters written by 26 outstanding scientists, including engineers and military scientists, among which we can find names as P. M. Pardalos, N. Uzunoglu, N. Limnios, C. Athanasiadis, N. J. Daras, M. Mitrouli, A. Burnetas, G. Kaimakamis, N. B. Zographopoulos, P. Michalis, V. Oikonomou, S. Tassopoulos and C. Koukouvinos, just to name a few.
The editor of the volume, Nicholas J. Daras (Hellenic Military Academy) has succeeded to bring together very essential research works as well as research expository essays of current international interest.